A POSTERIORI ERROR ESTIMATE OF FINITE ELEMENT METHOD FOR THE OPTIMAL CONTROL WITH THE STATIONARY BÉNARD PROBLEM

doi:10.4208/jcm.1210-m3864

Journal of Computational Mathematics ›› 2013, Vol. 31 ›› Issue (1): 68-87.DOI: 10.4208/jcm.1210-m3864

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A POSTERIORI ERROR ESTIMATE OF FINITE ELEMENT METHOD FOR THE OPTIMAL CONTROL WITH THE STATIONARY BÉNARD PROBLEM

Yanzhen Chang¹, Danping Yang²

1. Department of Mathematics, Beijing University of Chemical Technology, Beijing 100029, China;
2. Department of Mathematics, East China Normal University, Shanghai 200062, China

Received:2011-09-09 Revised:2012-10-25 Online:2013-01-15 Published:2013-01-17
Supported by:
This paper is supported in part by China NSF under the grant 11101025, the Fundamental Research Funds for the Central Universities and the Science and Technology Development Planning Project of Shandong Province under the grant 2011GGH20118.

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Yanzhen Chang, Danping Yang. A POSTERIORI ERROR ESTIMATE OF FINITE ELEMENT METHOD FOR THE OPTIMAL CONTROL WITH THE STATIONARY BÉNARD PROBLEM[J]. Journal of Computational Mathematics, 2013, 31(1): 68-87.

In this paper, we consider the adaptive finite element approximation for the distributed optimal control associated with the stationary Bénard problem under the pointwise control constraint. The states and co-states are approximated by polynomial functions of lowestorder mixed finite element space or piecewise linear functions and control is approximated by piecewise constant functions. We give the a posteriori error estimates for the control, the states and co-states.

Key words: Optimal control problem, Stationary Bé, nard problem, Nonlinear coupled system, A posteriori error estimate

CLC Number:

49J20
65N30

[1] F. Abergel and F. Casas, Some optimal control problems of multistate equations appearing in fluid mechanics, Math. Model. Numer. Anal., 27 (1993), 223-247.

[2] G. Alekseev, Solvability of stationary boundary control problems for heat convection equations, Sib. Math. J., 39 (1998), 844-858.

[3] N. Arada, E. Casas and F. Tröltzsch, Error estimates for the Numerical Approximation of a semilinear elliptic optimal control problem, Comput. Optim. Appl., 23 (2002), 201-229.

[4] F. Brezzi, J. Rappaz and P. Raviart, Finite-dimensional approximation of nonlinear problem. Part I: branches of nonsingular solutions, Numer. Math., 36 (1980), 1-25.

[5] E. Casas, Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, ESAIM Control, Optim. Cal. Var., 8 (2002), 345-374.

[6] E. Casas and F. Tröltzsch, Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory, SIAM J. Optim., 13 (2002), 406-431.

[7] E. Casas, F. Tröltzsch and A. Unger, Second order sufficient optimality conditions for some state constrained control problems of semilinear elliptic equations, SIAM J. Control Optim., 38 (2000), 1369-1391.

[8] Y. Chang and D. Yang, Superconvergence analysis of finite element methods for optimal control problems of the stationary B′enard type, J. Comp. Math., 26 (2008), 660-676.

[9] C. Cuvelier, Optimal Control of a System Governed by the Navier-Stokes Equations Coupled with the Heat equations, New Developments in Differential Equations (W. Eckhaus, ed.), Amsterdam: North-Holland, 1976.

[10] V. Girault and P. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, 1986.

[11] M. Gunzburger, L. Hou and T. Svobodny, Heating and cooling control of temperature distributions along boundaries of flow domains, J. Math. Syst. Estim. Control., 13 (1993), 147-172.

[12] M. Gunzburger and H. Lee, Analysis, approximation, and computation of a coupled solid/fluid temprature control problem, Comput. Methods Appl. Mech. Eng., 118 (1994), 133-152.

[13] A. Kufner, O. John and S. Fucik, Function spaces, Nordhoff, Leyden, The Netherlands, 1977.

[14] K. Kunusich, W. Liu, Y. Chang and N. Yan, R. Li, Adaptive finite element approximation for a class of parameter estimation problems. J. Comp. Math., 28 (2010), 645-675.

[15] H. Lee, Analysis of optimal control problems for the 2-D stationary Boussinesq equations, J. Math. Anal. Appl., 242 (2000), 191-211.

[16] H. Lee, Optimal control problems for the two dimensional Rayleigh-B′enard type convection by a gradient method, Japan. J. Indust. Appl. Math., 26 (2009), 93-121.

[17] H. Lee, Analysis and computations of Neumann boundary optimal control problems for the stationary Boussinesq equations, Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida USA, Dec., 2001.

[18] J. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.

[19] W. Liu and N. Yan, A posteriori error estimates for control problems governed by nonlinear elliptic equations, Appl. Numer. Math., 47 (2003), 173-187.

[20] P. Raviart and J. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, Springer-Verlag, New York, 606 (1977), 292-315.

[21] D. Yang and L. Wang, Two finite element schemes for steady convective heat transfer with system rotation and variable thermal properties, Numer. Heat Trans. B, 47 (2005), 343-360.

[22] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems I: a linear model problem, SIAM J. Numer. Anal., 28 (1991), 43-77.

[23] R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: basic concept, SIAM J. Control Optim., 39 (2000), 113-132.

[24] R. Verfürth, A posteriori error estimates for nonlinear problems, Math. Comp., 62 (1994), 445-475.

[25] R. Verfürth, A Review of Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Wiley-Teubner, New York, 1996.

[26] R. Li, W. Liu, H. Ma and T. Tang, Adaptive finite elememt approximation of elliptic optimal control, SIAM J. Control Optim., 41 (2002), 1321-1349.

[27] L. Scott and S. Zhang, Finite element interpolation of nonsmooth functions stisfying boundary conditions, Math. Comp., 54 (1990), 483-493.

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A POSTERIORI ERROR ESTIMATE OF FINITE ELEMENT METHOD FOR THE OPTIMAL CONTROL WITH THE STATIONARY BÉNARD PROBLEM

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